Curves of Constant Ratio with Quasi frame in E^3

Abstract

In the present study we handle a regular unit speed curve by means of the position vector given by the vectorial equation γ (s)=m0 t(s)+m1 nq (s)+m2 bq (s) where bq (s), nq (s) and t(s) are quasi frame vectors. Firstly, we analysis these curves and investigate to being constant ratio curve. Then, we give the parameterizations of T-constant and N- constant curve in accordance with quasi frame. Further, we get the conditions for a regular curve to correspond to be a W- curve in E^3.

Keywords

• Constant ratio curve
• Position vector
• Quasi frame

References

1. Chen BY. Constant ratio Hypersurfaces. Soochow J. Math. 2001; 28: 353-362.
2. Chen BY. Convolution of Riemannian manifolds and its applications. Bull. Aust. Math. Soc. 2002; 66: 177-191.
3. Chen BY. When does the position vector of a space curve always lies in its rectifying plane? Amer. Math. Monthly 2003; 110: 147-152.
4. Chen BY. More on convolution of Riemannian manifolds. Beitrage Algebra und Geom. 2003; 44: 9-24.
5. Chen BY and Dillen F. Rectifying curves as centrodes and extremal curves. Bull. Inst. Math. Acedemia Sinica 2005; 33: 77-90.
6. Ekici C., Göksel M. B., Dede M., Smarandache curves according to q − frame in Minkowski 3 − space, Conference Proceedings of Science and Technology, 2019; 2(2): 110 118.
7. Elshenhab A. M., Moazz O., Dassios I., Elsharkawy A., Motion along a space curve with a quasi frame in Euclidean 3 − space: Acceleration and Jerk, Symetry 2022; 14: 1610.
8. Ezentaş R and Türkay S. Helical versus of rectifying curves in Lorentzian spaces. Dumlıpınar Univ. Fen Bilim. Esti. Dergisi 2004; 6: 239-244.

9. Gray A. Modern differential geometry of curves and surface. USA: CRS Press Inc., 1993.
10. Gürpınar S, Arslan K, Öztürk G. A Characterization of Constant − ratio Curves in Euclidean 3 − space E3. Acta Universtatis Apulensis, Mathematics and Informatics. 2015; 44: 39-51.
11. Ilarslan K, Nesovic E and Petrovic TM. Some characterization of rectifying curves in the Minkowski 3 − space. Novi Sad J. Math. 2003; 32: 23-32.
12. Ilarslan K and Nesovic E. On rectifying curves as centrodes and extremal curves in the Minkowski 3 − space E3 1. Novi. Sad. J. Math. 2007; 37: 53-64.
13. Ilarslan K and Nesovic E. Some characterization of rectifying curves in the Euclidean space E4. Turk. J. Math. 2008; 32: 21-30.
14. Ilarslan K and Nesovic E. Some characterization of null, pseudo-null and partially null rectifying curves in Minkowski space − time. Taiwanese J. Math. 2008; 12: 1035-1044.
15. Klein F. and Lie S. Uber diejenigen ebenenen kurven welche durch ein geschlossenes system von einfach un endlich vielen vartauschbaren linearen Transformationen in sich ¨ ubergehen. Math. Ann. 1871; 4: 50-84.